## Engineering

1.0 Dependence of Relative Permeability on Petrophysical Properties

Carbon dioxide (CO2) entering the atmosphere from sources such as power plants, oil exploratory and production sites and traffic pollution from fossil fuel-burning vehicles. CO2 exists in high concentrations in the atmosphere which increases the impact of the greenhouse gas phenomena which makes the global warming problem worse. The relative permeability of soils is linked to finding a solution for removing large concentrations of CO2 by sequestering CO2 in appropriate geological formations. Data from a study by Bennion and Bachu (2005) is available online at the Relative Permeability Explorer. The Explorer is the home of a omputer model which allows users to understand how values of average porosity, relative permeability, pore volume, pore pressure and temperature effect the CO2 saturation and Brine saturation. This paper has attempted to offer data from models determining relative permeability in relationship to similar parameters presented in the Bennion and Bachu (2005) research.

Empirical parametric models for data fitting were studied. The first was the Brooks and Corey (1966) article titled ‘Properties of porous media affecting fluid flow.’ Secondly this report reviews a historical important trilogy of papers published during 1987 and 1988 by Parker, Lenard and Kuppusamy (1987) and Parker and Leonard (1987; 1988). The papers address the steps taken to address the scaling challenge which must be overcome in order to fit experimental data from a multi-phase flow in a porous media to real-world data. With an appropriate scaling model the understanding of the phenomena of relative permeability and its relationship to capillary-hydraulic properties (such as pressure) in the environment can be better understood. This is in turn can be used to better understand the optimum conditions for the sequestering CO2 in carbon dioxide absorbing geological formations.

1.1 Definitions

Capillary pressure can be defined as the Pressure of the more dense phase subtracted from the Pressure of the less dense phase.

Capillary Pressure – Permeability Relationship is a relationship that functions in saturation phases in a porous sample. If we assume a oil and water flow without a gaseous phase then permeability can be described as a function of saturation (S): kro(S) and krw(S). The capillary pressure (P) can also be described as a function of saturation: Pc(Sw).

Drainage refers to the case when saturation decreases in the wetting phase during a process occurring in porous media. For example in his research, Braun (1995: 223) used the term drainage to describe “decreases in water saturation for a water-oil system regardless of rock wettability.” Primary drainage relative permeability curves describe the situation when water saturation is decreasing from 100 percent. Secondary drainage curves “involve a decrease from the high water saturation occurring when immobile oil is present” (Braun 1995: 223).

Hysteresis in physics and the natural sciences refers to a lagging effect or retardation that occurs when forces acting on or against a body are changed. Viscosity and internal friction are two examples.

Permeability, absolute permeability and intrinsic permeability are terms describeing how much flow is capable of passing through a porous media. Darcy’s law is used to calculate high permeability or low permeability (the capacity of flow); the higher the permeability the higher the capacity for flow. Permeability is measured in units of darcies and expressed as length squared. Darcy’s Law assumes no gravitational effects. Permeability is dimensionless and ranges from 0 to 1.

. Eqn. 3

Equations 2 and 3 are both equations for multi-phase flow, they both define relative permeability flowing in x-direction. The subscript ‘o’ denotes oil. The subscript ‘w’ denotes water. The relative permeability for oil is denoted as ‘kro’ and the relative permeability for water is denoted as ‘krw’

The capillary pressure which is important in this paper’s discussion is defined by Equation 4; the difference in pressure between two phases

the subscript ‘c’ denotes the capillary pressure. Therefore Equation 4 is stating that the difference in pressure in the oil phase and the pressure in the water phase is equal to the capillary pressure.

Relative permeability hysteresis may occur or may not occur in fluid systems depending on the composition of the phases being observed. Braun (1995) reported that in a research project for building a conceptual model from laboratory measurements that relative permeability hysteresis evident in water phases but no hysteresis was evident in the oil phase. Relative permeability curves result from graphing data versus relative permeability in order to understand and predict processes in different fluid phases. Braun (1995: 222) explained that “In the case of relative permeability, two-phase flow properties of a porous medium depend on which phase is increasing in saturation.”

Saturation refers to the pore space fraction occupied by a phase; in a two phase system of oil and water Sw + So = 1.

The paper also reviews the development of modern computer models with reference to relative permeability and the capillary-hydraulic properties of flow through porous media. This was done in order to appreciate the earlier work done by modelers and how their work has laid the foundation for models such as the Bennion and Bachu model. Relative permeability in porous media means taking a detailed look at the structure of soils and at the spaces between grains of soil. The saturation of the media and the flow rate are two other relevant properties. The properties are discussed and their practical use in the models is explained and defined whenever possible.

2.0 Compile laboratory and field measurements of relative permeability in the literature

During April 1987 J.C. Parker, R. J. Lenard and T. Kuppusamy (1987) published the first of a trilogy of papers on identifying properties in that control multiphase flow by fitting experimental data to multi-phase flow in a porous media by developing a scaling model. The paper focused on the parametric modeling of multiphase flow in porous media based on the properties of the soil type. The purpose of the research described in the first paper was “to describe relative permeability-saturation-fluid pressure functional relationships in two- or three-fluid phase porous media systems subject to monotonic saturation paths” (Parker, Lenhard and Kuppusamy, 1987:618). Organic wastes that can cause a health risk when they reach groundwater aquifers which are used for drinking water. The reason the functional relationships have become particularly important is due to pollutants which enter water and move following convection dynamics but exist at different saturations in three phases of the environment: gas, water and as a nonaqueous phase liquid (NAPL).

Scaling is the term that can used to refer to fitting experimental data to real life situations. Scaling is a challenge that must be addressed when modeling flow through porous media when the “constitutive relationships governing multiphase contaminant movement” (Parker, Lenhard and Kuppusamy, 1987:618) such as permeabilities exist in different co-existent phases. The capillary-hydraulic properties of a porous media can be described by its relative permeability. Parker, Lenhard and Kuppusamy (1987) concluded that the model they created was successful in describing the permeabilities in two – phase systems. “Experimental results for two porous media with different two-phase fluid combinations indicate the scaling approach provides a good representation of two-phase saturation-pressure data” (Parker, Lenhard and Kuppusamy, 1987:623). The parameters use in the model were α, n, and Sr. where α and n are curve shape parameters in the following equation which represents the scaled saturation-capillary head relation. S represents the water functionalities of saturation, gaseous saturation (Sa), Sm which is an “apparent minimum or irreducible wetting fluid saturation” (Parker, Lenhard and Kuppusamy, 1987: 619). The wetting fluid saturation in each of two phases is described with the following equation

Figure 1 depicts two graphs the one on the left represents Saturation in the three phased of air-water, air-oil, and oil-water. The graph on the right represents the Effective Saturation in the three phases after scaling was performed.

On the left graph the wetting fluid Saturation is graphed on the x-axis and the unscaled Capillary Head (in cm H2O) is graphed on the y-axis. The medium is two-fluid air –water, air-NAPL, and NAPL-water systems. The soil media is sandy porous. The NAPL used for the experiment was p-cymene. The experiment arrived at each data point’s value by calculating the mean of triplicate trials. (Parker, Lenhard and Kuppusamy 1987: 619). The air-oil and oil-water phase curves overlap well in the region of 0 to 50 cm H2O. Whereas the air-water phase measurements do not overlap at any depth cm H2O with the other two phases. The air-water phase curve shifts approximately 0.2 units higher than for the other two phases which contain oil although the depth cm H2O has remained the same. (Parker, Lenhard and Kuppusamy 1987: 619 ) In the graph on the right in Figure 1 the one curve represents the effective saturation (graphed on the x-axis) when the three phases as represented in Figure 1 right are replaced by a curve of the scaled saturation-pressure function; the scaled capillary head data points are measured in cm and graphed on the y-axis. The top of the curve is higher by 60 cm and this time the curve has shifted upwards along the y-axis.

Figure 1 Comparison of Saturation when graphed versus the Capillary Head and the Scaled Capillary Head (Parker, Lenhard and Kuppusamy, 1987: 619-620)

The scaled saturation-capillary head function (Figure 1 right) was considered an index of the pore size and distribution of the sandy porous soil media. The researchers noted that the assumption that was the most difficult to contend with in the model was that “of a rigid porous medim with no significant fluid-solid phase interactions” (Parker, Lenhard and Kuppusamy, 1987: 623).

The paper published in December 1987 by J. C. Parker and R. J. Lenhard is titled “A Model for Hysteretic Constitutive Relations Governing Multiphase Flow 1. Saturation-Relations.” The model designed was based on the interaction between saturation and pressure with a discussion of both two-phase and three-phase porous media systems. The model the researchers developed was used to predict “arbitrary order scanning curves” using (a) an empirical interpolation and (b) a scaling procedure (Parker and Lenhard 1987: 2187).The data use in the model defined the path through p-Cymene-Air, Air-Water and p-Cymene systems in a sandy porous medium by measuring h (water height equivalent) and S (saturation) in three replicates. Each system’s data was presented in two sections from (a) the main drainage and (b) the Saturation-Capillary Head.

The third paper published in 1988 by Lenhard and Parker was titled “Experimental validation of the theory of extending two-phase saturation-pressure relations to three-fluid phase systems for monotonic drainage paths.” The paper describes the resulting model which the authors concluded could describe Saturation-Pressure relationships in 2- or 3-fluid phase systems. The model was developed from the fir two papers and uses the same notation as well as the same soil medium of sandy porous. The Particle size distributions and packed bulk densities used by Lenhard and Parker (1988) can be found in Table 4.

The scaling method described above from Parker (et al. 1987) was incorporated in the final model. An appropriate parametric depiction of the scaled retention function was developed under the scenario of a multi-phase retention function so that saturation-pressure relationships could be satisfactorily described in both two- and three-fluid phases. The researchers concluded that

“the results indicate that calibration of the model by fitting to two-phase air-water, air-oil, and oil-water S-P data should yield an accurate description of two- and three-phase drainage S-P relations. If two-phase air-oil and oil-water S-P relations are not available, model parameters can be estimated from two-phase air-water S-P data and relatively simple measurements of air-water, air-oil, and oil-water interfacial tensions with only moderate deterioration in model precision.” (Lenhard and Parker 1988: 379).

Marshall (1955) derived a scaled relationship between permeability and the size distribution of pores by adapting the Kozney equation. The research was conducted using an isotropic material . Marshal (1955) derived an equation by integrating Darcy’s Law to describe “the flow of fluid through a capillary tube, the permeability of a model porous material made up of parallel tubes of uniform size could be expressed as a function of its porosity and pore radius” in the following equation (Marshall, Holmes and Rose 1958: 212)

Assumption 2: “All model parameters are defined in terms of measurements which may be obtained from two-phase systems (air-water, air-oil, oil-water). Extension to three-phase systems is based on the assumption that fluid entrapment processes in three phase systems are similar to those in two-phase systems and that wettability decreases in the order: water to oil to air” (Parker and Lenhard, 1987: 2187).

k =relative permeability

S =fluid saturation

P = pressure

h -= height of fluid phase

Subscripts used in the model describe the type of fluid phase present in the porous medium a represents air; o represents oil and w represents water.

Saturations of the sandy porous soil was determined by taking measurements of Sa (ha, ho, hw), So (ha, ho, hw), and Sw (ha, ho, hw) by “draining the oil and water from the total liquid saturated condition” (Parker and Lenhard, 1988: 376).

The following symbols and definitions describe as follows hij= hi - hj

when hi = Pi/pw g and hj = Pj/pw g where Pi and Pj are fluid pressures of nonwetting phase i and wetting phase Pj (i,j = a, ,o, w). pw is equal to the water density. g is the gravitational acceleration.

2.2.2 From McGuire, Weiler and McDonnel (2007)

n = average soil porosity

The drainable porosity (nd) is defined by the difference in volumetric water content between 0 and 100 cm of water potential (i.e., approximately from saturation to field capacity).

3.0 Prediction of Relative Permeability (Bennion and Bach 2005)

The average porosity for the 17 sample types in the Bennion and Bachu (2005) research is equal to 0.145. (See table 5). Note that four of the sample types showed no permeability data. Those were the Wabamun Carbonate (High K, Fresh Water), Frio Sandstone, Cardium Sandstone (IFT = 56.2 mN/m), Cardium Sandstone (IFT = 32.2 mN/m), and Cardium Sandstone (IFT = 19.8 mN/m). Therefore calculating the average porosity for the 15 remaining samples gives 0.154 as the average porosity; the lowest value was 0.079 for Wabamun Carbonate (Low K) and the highest value was 0.26 for Frio sandstone. Frio sandstone has no permeability data.

For Permeability the 12 of the samples offered values ranging from the minimum 0.21 mD Wabamun Carbonate (Low K; k = potassium) to the maximum 645 mD the Cooking Lake Carbonate. The average permeability is 201.728 mD.

Wabamun Carbonate (Low K) shows brine drainage with a CO2 saturation of 0.041 and relative permeability of 0.8614. In the carbon dioxide drainage the CO2 saturation 0.405 and the relative permeability is 0.5289. Therefore the brine drainage is very similar in terms of CO2 saturation and the relative permeability is higher by 0.3325 mD.

Cooking Lake Carbonate in the brine drainage scenario shows zero CO2 saturation but a relative permeability of 1. Cooking Lake Carbonate in the carbon dioxide drainage scenario shows 0.524 for CO2 saturation but a relative permeability of only 0.069.

The above discussion has focused on the theories that have been the foundations of models representing relative permeability. The development of equations from one researcher to the next as new ideas were added to the models shows the growth of the development of the understanding of how soil physics works by using arithmetic and calculus to create models. The discussion brings together some of the earliest models of soils relative permeability and the relationship to capillary-hydraulic properties (in particular soil structures). Two contemporary models were studied. The Bennion and Bachu (2005) can be used at the Relative Permeability Explorer site. The second model studied was from the work of McGuire, Weiler and McConnell (2007) which was interesting because the researchers integrated tracer experiments with computer models in order to deduce water residence times.

The contemporary model developed by Bennion and Bachu (2005) is available from the Internet as the Relative Permeability Explorer. The different properties of several soils can be explored by changing the parameters graphed. The purpose of the Bennion and Bachu (2005) is to use the property of relative permeability in the search for the best places to do CO2 sequestering in the United States. The Relative Permeability Explorer model demonstrated that the soil type Cooking Lake Carbonate, in the brine drainage scenario shows zero CO2 saturation and a relative permeability of 1. Therefore sites with Cooking Lake Carbonate can be assumed to be good locations for sequestering of atmospheric CO2.

The early models that were reviewed, analyzed and their parameters defined in included a model which was published in the literature in 1958 by T.J. Marshall and was a major part of his book (with co-authors Holmes and Rose and first published in 1979). Marshall (1958) used modelling to describe the relationship between permeability and the size distribution of pores. Lenhard and Parker and Kuppusamy (1987) published the first in a trilogy of papers developing a parametric model which would eventually describe the properties that govern two- and three-phase flow in porous media. Brooks and Corey (1966) developed the same type of model even further by modelling the porous media properties that affect fluid flow.

Data from the Bennion and Bachu Relative Permeability Explorer was discussed and used to examine the relationship between permeability and CO2 sequestering. The earlier models reviewed are foundational models that relate relative permeability with soil moisture. The contemporary models use the same type of parametric models and their derivations for defining the relationship between permeability and CO2 sequestration.

## References

Benson, S. (n.d.). Relative Permeability Explorer [Online] Benson Lab, Available from <https://pangea.stanford.edu/research/bensonlab/relperm/index.html> [10 Oct. 2013]

Bennion, B., Bachu, S. (2005). Relative Permeability Characteristics for Supercritical CO2 Displacing Water in a Variety of Potential Sequestration Zones in the Western Canada Sedimentary Basin. [Online] Paper SPE 95547, presented at the 2005 SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, October 9-12, 2005. Available from <https://pangea.stanford.edu/research/bensonlab/relperm/index.html> [20 Oct. 2013]

Bennion, D.B. and Bachu, S. 2008. Drainage and Imbibition Relative Permeability Relationships for Supercritical CO2/Brine and H2S/Brine Systems in Intergranular Sandstone, Carbonate, Shale, and Anhydrite Rocks. SPE Res Eval & Eng, 11 (3): 487-496. SPE-99326-PA.

Braun, E.M. (1995). ‘Relative Permeability Hysteresis: Laboratory Measurements and a Conceptual Model.’ SPE Reservoir Engineering, 10(3), 222-228.

Brooks, R.H. and Corey, A.T (1966). ‘Properties of porous media affecting fluid flow’, Journal of Irrigation and Drainage Division of the American Society of Civil Engineers, 92, 61-88.

Lenhard, R.J. and Parker, J.C. (1988). ‘Experimental validation of the theory of extending two-phase saturation-pressure relations to three-fluid phase systems for monotonic Drainage Paths.’ Water Resources Research, 24(3), 373-380.

Marshall, T.J. (1958). ‘A relation between permeability and size distribution of pores.’ Journal of Soil Sciences, 9(1), B1-B8.

Marshall, T.J., Holmes, J.W. and Rose, C.W. (1999) Soil Physics, 3rd ed., Cambridge: Cambridge University Press

McGuire, K.J., Weiler, M., and McDonnell, J.J. (2007). ‘Integrating tracer experiments with modelling to assess runoff processes and water transit times.’ Water Resources, 30, 824-837)

Parker, J. C., Lenhard, R. J. and Kuppusamy, T. (1987).’ A parametric model for constitutive properties governing multiphase flow in porous media’, Water Resources Research 23(4), 618-624.

Parker, J.C. and Lenhard, R.J. (1987). ‘A model for Hysteretic Constitutive Relations Governing Multiphase Flow 1. Saturation-Pressure Relations.’ Water Resources Research, 23(12), 2187-2196.

Lenhard, R.J. and Parker, J.C. (1988). ‘Experimental validation of the theory of extending two-phase saturation-pressure relations to three-fluid phase systems for monotonic drainage paths.’ Water Resources Research, 24(3), 373-380.